/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 17 15.17 By confounding \(A B C\) i... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 15: Problem 17

15.17 By confounding \(A B C\) in two replicates and \(A B\) in the third, show the block arrangement and the analysis-of-variance table for a \(2^{3}\) factorial experiment with three replicates. What is the relative information on the confounded effects?

Short Answer

Expert verified
The relative information about the confounded effects of \(ABC\) in the first two replicates and \(AB\) in the third replicate is embedded in the block variance and can be difficult to distinguish. However, it is still possible to get an overall picture of the major factor effects. This outcome will depend on the observed data, as well as the resulting F0 and p-values in the analysis-of-variance table.

Step by step solution

01

Setup the Block Arrangement for the First Two Replicates

Console with the confounding pattern of \(ABC\). For two replicates of the \(2^{3}\) factorial design, we have \(2^{3} = 8\) experimental conditions: \(000, 001, 010, 011, 100, 101, 110\) and \(111\), with three factors \(A, B\) and \(C\). We divide these eight experimental conditions into two blocks according to the confounding pattern \(ABC\). Block I is \(000, 001, 010, 011\) and Block II is \(100, 101, 110, 111\).
02

Setup the Block Arrangement for the Third Replicate

For the third replicate, apply the confounding pattern \(AB\). The block arrangement becomes: Block I: \(000, 001, 100, 101\) and Block II: \(010, 011, 110, 111\).
03

Set up Analysis-of-Variance table

Set up a table with columns for source of variation (blocks, treatments, error, total), degrees of freedom, sum of squares (SS), mean sum of squares (MSS), and F0 (variance ratio). Remember to split the treatments column into A, B, C, AB, AC, BC, ABC. The degrees of freedom will add up to a total of 23 (24 observations - 1).
04

Calculate the Sum of Squares

Once the data values are obtained, calculate the sum of squares for each source of variation. Remember that SS for total equal SS for treatments + SS for blocks + SS for errors. After, divide each SS by its corresponding degrees of freedom to get the mean sum of squares.
05

Calculating F0

Divide MSS for each treatment variation by MSS for the error to get the F0 values. These are used to test the significance of each factor effect.
06

Discuss Relative Information on Confounded Effects

With confounding for AB and ABC, the relative information on these effects is mixed in with the block variance in the analysis of variance table. It means that the information about these effects is diluted and it is harder to distinguish their individual contributions. However, it saves experimental resources and may still present an overall picture of the major factor effects. In the presented scenario, depending on the F0 and p-values for ABC and AB, we might be able to differentiate their effects.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factorial Experiment
A factorial experiment is a type of experimental design where investigators study the effects of two or more factors simultaneously. This approach allows for an understanding of how different factors interact and influence outcomes.
Factorial experiments are particularly effective in revealing interactions and main effects of factors. Consider a scenario with three factors, denoted by A, B, and C, each with two levels. It implies a total of 2^3 = 8 experimental conditions. Each combination of factor levels represents an experimental condition, such as 000, 001, ... 111, where each digit represents the level of one factor (0 or 1).
  • Main Effects: These refer to the influence of an individual factor, averaged over the levels of other factors.
  • Interaction Effects: These describe how the effect of one factor changes across the levels of another factor.
Factorial experiments offer a comprehensive framework to study complex interactions in a controlled manner, minimizing the number of experiments required to understand multiple factors and their interactions.
Analysis of Variance (ANOVA)
The Analysis of Variance (ANOVA) is a statistical method used to determine whether there are any statistically significant differences between the means of more than two groups. It assesses if observed differences are likely due to experimental treatment or random variation among the groups.
ANOVA tables are used to break down the variance observed in the data into components:
  • Sources of Variation: Includes blocks, treatments, and error. It helps identify where the variance originates.
  • Degrees of Freedom (DF): Indicates the number of independent values or quantities involved in the calculations.
  • Sum of Squares (SS): Measures the total variation among the data points.
  • Mean Sum of Squares (MSS): Calculated by dividing SS by the respective degrees of freedom.
  • F-ratio (F0): Obtained by dividing the MSS of treatments by the MSS of errors. It is used to test the hypothesis that the treatment means are equal.
In factorial designs, ANOVA helps ascertain the significance of each factor and their interactions. While performing ANOVA, separating treatment effects from noise (error) ensures efficient experiment interpretation.
Blocking in Experiments
Blocking is a technique used to control extraneous variability in experiments. In essence, it is about grouping experimental units into blocks that are similar to each other. This division helps isolate the variability owing to known confounding factors so that it does not affect the factor being studied.
Consider the context where a factorial experiment is being implemented. By dividing the trials into blocks, it becomes feasible to examine the impact of primary factors more precisely. For instance:
  • Replicates and Confounding: Confounding is a practice where certain effects are designated as indistinguishable from the blocks. This results in a more straightforward experiment but at the cost of not precisely identifying some interactions.
  • Reduced Variation: Control for variability by having similar conditions within each block, thus improving the reliability of the findings.
It's common to confound higher-order interactions, like ABC, within the blocks in a factorial experiment. Although this means detailed study on these specific confounded effects is limited, the overall robustness of evaluating the primary factors is strengthened, conserving resources and time. This strategy performs well in complex experimental scenarios.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In Myers and Montgomery (2002), a data set. is discussed in which a \(2^{3}\) factorial is used by an engineer to study the effects of cutting speed \((A)\), tool geometry (B). and cutting angle \((C)\) on the life (in hours) of a machine tool. Two levels of each factor are chosen, and duplicates were run at, each design point with the: order of the runs being random. The data are presented here. (a) Calculate all seven effects. Which appear, based on their magnitude, to be important? (b) Do an analysis of variance and observe \(P\) -values. (c) Do your results in (a) and (b) agree? (d) The engineer felt confident that cutting speed and cutting angle should interact. If this interaction is significant, draw an interaction plot and discuss the engineering meaning of the interaction. $$ \begin{array}{ccccc} & A & B & C & \text { Life } \\ \hline(1) & \- & \- & \- & 22.31 \\ a & \+ & \- & \- & 32,43 \\ b & \- & -i & \- & 35.34 \\ a b & \+ & \+ & \- & 35.47 \\ c & \- & \- & \+ & 44,45 \\ a c & \+ & \- & \- & 40.37 \\ b c & \- & \+ & \+ & 60,50 \\ a b c & \- & \- & \+ & 39,41 \end{array} $$

In a study Durability of Rubber to Steel Adhesively Bonded Joints conducted at. the Department of Environmental Science and Mechanics and analyzed by the Statistics Consulting Center at the Virginia Polytechnic Institute and State University, an experimenter measures the number of breakdowns in an adhesive seal. It was postulated that concentration of seawater \(A\), temperature \(B\), \(\mathrm{pH} C\), voltage \(\mathrm{D}\), and stress \(E\) influence the breakdown of an adhesive seal. A \(\frac{1}{2}\) fraction of a \(2^{\prime \prime}\) factorial experiment is used, with the defining contrast being \(A B C D E\). The data are as follows: $$ \begin{array}{rrrrrr} A & B & C & D & E & \text { Response } \\ \hline-1 & -1 & -1 & -1 & 1 & 462 \\ 1 & -1 & -1 & -1 & -1 & 746 \\ -1 & 1 & -1 & -1 & -1 & 714 \\ 1 & 1 & -1 & -1 & 1 & 1070 \\ -1 & -1 & 1 & -1 & -1 & 474 \\ 1 & -1 & 1 & -1 & 1 & 832 \\ -1 & 1 & 1 & -1 & 1 & 764 \\ 1 & 1 & 1 & -1 & -1 & 1087 \\ -1 & -1 & -1 & 1 & -1 & 522 \\ 1 & -1 & -1 & 1 & 1 & 854 \\ -1 & 1 & -1 & 1 & 1 & 773 \\ 1 & 1 & -1 & 1 & -1 & 1068 \\ -1 & -1 & 1 & 1 & 1 & 572 \\ 1 & -1 & 1 & 1 & -1 & 831 \\ -1 & 1 & 1 & 1 & -1 & 819 \\ 1 & 1 & 1 & 1 & 1 & 1104 \end{array} $$ Perform an analysis of variance on main effects, and two-factor interactions. assuming that all three-factor and higher interactions are negligible. Use \(\mathrm{ev}=0.05\).

A \(2^{2}\) factorial experiment is analyzed by the Statistics Consulting Center at Virginia Polytechnic Institute and State University. The client is a member of the Department of Housing, Interior Design, and Resource Management. The client is interested in comparing cold start versus preheating ovens in terms of total energy being delivered to the product. In addition, the conditions of convection are being compared to regular mode. Four experimental runs were made at each of the four factor combinations. Following are the data from the experiment: $$ \begin{array}{lcl|cc} & \ {\text { Preheat }} & & {\text { Cold }} \\ \hline \text { Convection } & 618 & 619.3 & 575 & 573.7 \\ \text { Mode } & 629 & 611 & 574 & 572 \\ \hline \text { Regular } & 581 & 585.7 & 558 & 562 \\ \text { Mode } & 581 & 595 & 562 & 566 \end{array} $$ Do an analysis of variance to study main effects and interaction. Draw conclusions.

In an experiment conducted at the Department of Mechanical Engineering and analyzed by the Statistics Consulting Center at the Virginia Polytechnic Institute and State University, a sensor detects an electrical charge each time a turbine blade makes one; rotation. The sensor then measures the amplitude of the electrical current. Six factors arc rprn \(A\), temperature \(B\), gap between blades \(C\), gap between blade and casing \(D\), location of input \(E\), and location of detection \(F .\) A \(\frac{1}{4}\) fraction of a \(2^{\circ}\) factorial experiment is used, with defining contrasts being \(A B C E\) and \(B C D F .\) The data are as follows: $$ \begin{array}{rrrrrrr} A & B & C & D & E & F & \text { Response } \\ \hline-1 & -1 & -1 & -1 & -1 & -1 & 3.89 \\ 1 & -1 & -1 & -\mathrm{i} & 1 & -1 & 10.46 \\ -1 & 1 & -1 & -1 & 1 & 1 & 25.98 \\ 1 & 1 & -1 & -1 & -1 & 1 & 39.88 \\ -1 & -1 & 1 & -1 & 1 & 1 & 61.88 \\ 1 & -1 & 1 & -1 & -1 & 1 & 3.22 \\ -1 & 1 & 1 & -1 & -1 & -1 & 8.94 \\ 1 & 1 & 1 & -1 & 1 & -1 & 20.29 \\ -1 & -1 & -1 & 1 & -1 & 1 & 32.07 \\ 1 & -1 & -1 & 1 & 1 & 1 & 50.76 \\ -1 & 1 & -1 & 1 & 1 & -1 & 2.80 \\ 1 & 1 & -1 & 1 & -1 & -1 & 8.15 \\ -1 & -1 & 1 & 1 & 1 & -1 & 16.80 \\ 1 & -1 & 1 & 1 & -1 & -1 & 25.47 \\ -1 & 1 & 1 & 1 & -1 & 1 & 44.44 \\ 1 & 1 & 1 & 1 & 1 & 1 & 2.45 \end{array} $$ Perform an analysis of variance on main effects, and two-factor interactions. assuming that all three-factor and higher interactions are negligible. Use \(\alpha=0.05\).

In the study An X-Ray Fluorescence Method for Analyzing Polybutadiene-Acrylic Acid (PBAA) Propellants, Quarterly Reports, RK-TR-62-1, Army Ordnance Missile Command, an experiment was conducted to determine whether or not there is a significant difference in the amount of aluminum achieved in the analysis between certain levels of certain processing variables. The data given in the table were recorded. $$ \begin{array}{cccccc} &\ {\text { Phys. Mixing Blade Nitrogen }} \\ \text { Obs.} &{ State } & \text { Time } &\text { Speed } & {Condition}& { Aluminum}{ } \\ \hline \mathbf{1} & 1 & 1 & 2 & 2 & 16.3 \\ 2 & 1 & 2 & 2 & 2 & 16.0 \\ 3 & 1 & 1 & 1 & 1 & 16.2 \\ 4 & 1 & 2 & 1 & 2 & 16.1 \\ 5 & 1 & 1 & 1 & 2 & 16.0 \\ 6 & 1 & 2 & 1 & 1 & 16.0 \\ 7 & 1 & 2 & 2 & 1 & 15.5 \\ 8 & 1 & 1 & 2 & 1 & 15.9 \\ 9 & 2 & 1 & 2 & 2 & 16.7 \\ 10 & 2 & 2 & 2 & 2 & 16.1 \\ 11 & 2 & 1 & 1 & 1 & 16.3 \\ 12 & 2 & 2 & 1 & 2 & 15.8 \\ 13 & 2 & 1 & 1 & 2 & 15.9 \\ 14 & 2 & 2 & 1 & 1 & 15.9 \\ 15 & 2 & 2 & 2 & 1 & 15.6 \\ 16 & 2 & 1 & 2 & 1 & 15.8 \end{array} $$ The variables are given below. .4: mixing time level \(1-2\) hours level \(2-4\) hoursi B: \(\quad\) blade speed \(\begin{array}{lll}\text { level } & 1-36 & \text { rpin }\end{array}\) level \(2-78\) rpm \(C: \quad\) condition of nitrogen passed over propellant level \(1-\mathrm{dry}\) level \(2-72 \%\) relative humidity \(D: \quad\) physical state of propellant level 1 -uncured level 2 -cured Assuming all three- and four-factor interactions to be negligible, analyze the data. Use a 0.05 level of significance. Write a brief report summarizing the findings.
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.